Relativistic pendulum and invariant curves

TL;DR

This paper uses advanced mathematical theories to prove that solutions to the relativistic pendulum have bounded momentum and identifies quasiperiodic solutions, enhancing understanding of its long-term behavior.

Contribution

It applies KAM and Aubry-Mather theories to the relativistic pendulum, demonstrating bounded solutions and quasiperiodic behavior, which are novel insights in this context.

Findings

All solutions have bounded momentum

Existence of quasiperiodic solutions in a generalized sense

Application of modified Aubry-Mather theory to relativistic systems

Abstract

We apply KAM theory to the equation of the forced relativistic pendulum to prove that all the solutions have bounded momentum. Subsequently, we detect the existence of quasiperiodic solutions in a generalized sense. This is achieved using a modified version of the Aubry-Mather theory for compositions of twist maps.